Method for resistivity well logging utilizing nuclear magnetic resonance

ABSTRACT

A method for determining petrophysical properties of geological structures using nuclear magnetic resonance utilizes a well logging apparatus and estimates resistivity in the formation surrounding a borehole by measuring current field strength and distribution.

CROSS-REFERENCE TO RELATED PATENT APPLICATIONS

This application is a Division of U.S. application Ser. No. 09/567,117,filed May 8, 2000 (U.S. Pat. No. 3,342,784) incorporated herein byreference in its entirety, which is a Division of U.S. application Ser.No. 08/885,925, filed Jun. 30, 1997 (U.S. Pat. No. 6,166,540),incorporated herein by reference in its entirety.

FIELD OF THE INVENTION

The present invention relates to the determination of petrophysicalproperties of geologic structures and, more particularly, to resistivitywell logging using nuclear magnetic resonance.

DESCRIPTION OF THE RELATED ART

Measurement of rock formation resistivity by electrode well logging isan established technique for estimating the petrophysical properties ofthe formation surrounding a “conducting” borehole. Numerous devicesemploying various arrays of electrodes are, or have been, in use sincethe first resistivity log was recorded in Pechelbronn, France in 1927 byConrad Schlumberger. All such devices have in common the production of acurrent density field {overscore (J)} in the formation by an electricpower source and the mapping of potential differences along the boreholeor the mapping of electrode potentials required to maintain a specifiedcurrent distribution in the borehole. See, Bassiouni, Zaki, “Theory,Measurement, and Interpretation of Well Logs”; Society of PetroleumEngineers; Richardson, Tex., 1994, Chapter 5. All prior art electrodedevices measure voltages or currents at the internal surface of theborehole. Subsequent analysis is then performed to loosely correlatethese borehole measurements with some of the petrophysicalcharacteristics of the surrounding formation.

Induction tools were introduced in the mid-1940's to estimateresistivity in non-conducting boreholes. These devices magneticallyinduce a current flux in the formation surrounding the borehole, whichformation acts as a lossy distributed mutual inductance between two ormore measuring inductances. Exemplary of such a device is that shown anddescribed in U.S. Pat. No. 5,428,293 to Sinclair et al.

While such electrode and induction tools have been widely used over theyears, they have not proven to be fully satisfactory because theyprovide only gross approximations of resistivity distributors. Attemptshave been made to overcome some of the disadvantages of both inductionand of direct contact electrode current and voltage measurement devicesby using other logging techniques.

Nuclear magnetic resonance devices measure other related characteristicsof the rock formation surrounding a borehole. Nuclear magnetic resonancedevices have been applied, for example, to measure such geophysicalproperties as porosity, pore size distribution, bulk fluid volume, andirreducible bound fluid volume of geological formations surrounding aborehole. Applications of this type are exemplified in U.S. Pat. No.4,933,638 to Kenyon; U.S. Pat. No. 5,212,447 to Paltiel; U.S. Pat. No.5,280,243 to Miller; U.S. Pat. No. 5,389,877 to Sezginer; U.S. Pat. No.5,412,320 to Coates; U.S. Pat. No. 5,432,446 to Macinnis; U.S. Pat. No.5,486,761 to Freedman; and U.S. Pat. No. 5,557,200 to Coates.

Representative of magnetic resonance logging tools is a device marketedunder the mark MRIL by Numar Corporation. The Numar device has asensitive volume approximating a thin cylinder 24 inches in height, 16inches in diameter, and of one millimeter slice thickness surrounding aborehole of 8 to 12 inches diameter. This device permits measurements tobe made peripheral to the borehole mud, the mudcake on the borehole walland often to the flushed zone and the transition zone, yielding animproved estimate of the properties of the uninvaded formationsurrounding the borehole relatively free of borehole effect. See,Bassiouni; op. cit. p. 71, 72.

While the value of concordant resistivity data has long beenappreciated, and while the application of nuclear magnetic resonancetechniques to the derivation of reasonably accurate information on poresize, bulk fluid volume and similar physical characteristics of geologicformations has been recognized, the advantages attendant to thesimultaneous use of nuclear magnetic resonance for additionallydetermining the resistivity of geologic structures surrounding aborehole have not heretofore been realized.

SUMMARY OF THE INVENTION

Accordingly, it is an object of the present invention to provide in anembodiment a resistivity well logging method utilizing nuclear magneticresonance.

A further object of the present invention is to provide, in anembodiment, a method for estimating resistivity in a formationsurrounding a borehole utilizing magnetic resonance. The method includesproviding a magnetic resonance well logging tool with a Faraday shield,and placing the magnetic resonance well logging tool and Faraday shieldin the borehole and capacitively or conductively coupling the same withthe formation surrounding the borehole. The method further includesenergizing the formation surrounding the borehole by selectivelyapplying phase modulating voltages to the Faraday shield and an upperand a lower coupling member to establish current fields (i) in axial andradial directions, respectively, in the formation surrounding theborehole, and/or (ii) in at least one of axial and radial directions inthe formation surrounding the borehole. The method further includesmeasuring the current strength and distribution of the current fields bydetecting phase modulation of spins of materials in the sensitive volumeof the magnetic resonance tool.

It is also an object of the present invention to provide, in anembodiment, a method for determining resistivity in a formationsurrounding a borehole. The method includes capacitive or conductivecoupling a phase-modulating current to the formation by placing first,second, and third spaced coupling members in the borehole andselectively applying low frequency power signals to the first and secondcoupling members and to the second and third coupling members,respectively. The method further includes placing a well logging tool inthe borehole in proximity to the second coupling member.

Another object of the present invention is to provide, in an embodiment,another method for determining resistivity in a formation surrounding aborehole. The method includes placing capacitive coupling members aboveand below a center capacitive coupling member containing a magneticresonance well logging tool, and selectively establishing current fields(i) in axial and radial directions, respectively, in a sensitive regionaround the tool, and/or (ii) in at least one of axial and radialdirections in a sensitive region around the tool. The method furtherincludes determining resistivity in the sensitive region around the toolutilizing the current fields.

It is yet a further object of the present invention to provide, in anembodiment, yet another method for determining resistivity in aformation surrounding a borehole. The method includes placing a magneticresonance well logging tool in the borehole and energizing the tool toproduce a magnetic resonance sensitive volume thereabout. The methodfurther includes selectively measuring current flow in the formation (i)both perpendicular to and parallel to the borehole axis within andadjacent to the sensitive volume, and/or (ii) wherein the current flowis at least one of perpendicular to and parallel to the borehole axiswithin and adjacent to the sensitive volume. The method further includesdetermining resistivity in the formation utilizing the current flowwhich is measured.

Another object of the present invention is to provide, in an embodiment,a further method for determining resistivity in a formation surroundinga borehole. The method includes placing a magnetic resonance welllogging tool having a Faraday shield in the borehole and capacitively orconductively coupling the same with the formation surrounding theborehole. The method further includes placing upper and lower couplingmembers in the borehole above and below the Faraday shield of themagnetic resonance well logging tool, respectively, and capacitively orconductively coupling the same with the formation surrounding theborehole. The method further includes selectively applying voltages tothe Faraday shield and the upper and lower coupling members to establishcurrent fields (i) in axial and radial directions, respectively, in theformation, and/or (ii) in at least one of axial and radial directions inthe formation. The method further includes measuring the resultingnuclear magnetic resonance signals to determine resistivity within theformation.

A further object of the present invention is to provide, in anembodiment, a method for measuring diffusion coefficient and spinrelaxation time in a nuclear magnetic resonance well logging system. Themethod includes placing a nuclear magnetic resonance well logging devicein a borehole having a surrounding formation, and energizing the nuclearmagnetic resonance well logging device to generate a phase modulatingcurrent at a selected frequency and with a predetermined magneticresonance pulse interval. The method further includes varying theintensity of the phase modulating current and its frequency and themagnetic resonance pulse interval so as to change the amplitude of amagnetic resonance signal, and determining a value for diffusioncoefficient and a value for spin relaxation time utilizing the magneticresonance signal.

The methodology according to an embodiment of the present invention ofresistivity well logging may be used with any suitable well loggingtool, but is particularly well suited for use with magnetic resonancetools since magnetic field gradients are produced by the application oftime-varying electric fields as disclosed in applicant's U.S. Pat. No.5,412,322, entitled “Apparatus and Method for Spatially Ordered PhaseEncoding and for Determining Complex Permittivity in Magnetic Resonanceby Using Superimposed Time-Varying Electric Fields,” which isincorporated herein by reference.

The data generated by the magnetic resonance tool in accordance with anembodiment of this invention may be obtained by using any suitableoutput recovery technique, but is most suitably retrieved by theimplementation of techniques of demodulation and detection ofphase-modulated magnetic resonance signals, as more fully disclosedherein.

In the implementation of an embodiment of the present invention,electrode arrays supplied by periodic voltage or periodic currentsources are placed in either “conducting” or “non-conducting” boreholesand are configured to produce either predominately radial orpredominately axial periodic current fields. These fields are used tophase modulate signals produced by spin distributions created by anuclear magnetic resonance well logging device. Measurements may be madein low conductivity boreholes by capacitively coupling Very LowFrequency currents (e.g. 1-10 KHz) to the conductive formationsurrounding the borehole through impedance matching circuits.

A phase-modulated magnetic resonance signal consisting of a linespectrum is created by these current fields which can then be“demodulated” by convolution and cross-correlation with the master radiofrequency oscillator frequency H₁. Individual sideband “lines” can be“detected” by convolution and cross-correlation with integral multiplesof the phase-modulating frequency of the current field, all of which isthe radio engineering embodiment of statistically determining theamplitude of low power hidden periodicities in a stationary randomprocess when the frequency of these periodicities is a known integralmultiple of a reference signal available from a master oscillator. Thecentral and the peripheral components of apparent formation resistivityR_(a) may be estimated separately. Spin-spin relaxation T₂ and diffusionD may be estimated in the bulk non-surface associated large porecomponent of formation fluid.

The invention can also be used in measuring properties of samples in alaboratory setting, as well as in situ logging-type includinglogging/measuring while drilling (LWD/MWD) measurements.

Additional objects, advantages and features of embodiment of the presentinvention will become apparent to those skilled in the art from thefollowing description of the preferred embodiment when taken inconjunction with the attached appendices and accompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a partially pictorial, partially block diagram of a preferredembodiment of a resistivity well logging apparatus using nuclearmagnetic resonance in accordance with the present invention;

FIG. 2 is a partially pictorial, partially block diagram of thecomponents of the apparatus of FIG. 1 in accordance with the presentinvention;

FIG. 3 is a cross-sectional view of a coupling element and integraltransformer of the apparatus of FIG. 1;

FIG. 4 is a one-quarter cross-sectional view of the coupling element andintegral transformer taken along line 4—4 of FIG. 3;

FIG. 5 is a block diagram of the sideband detector of FIG. 2;

FIG. 6 is a schematic diagram of the phase-modulating power supply ofFIG. 2;

FIG. 7 is a diagrammatic view of the electrical interconnection of theupper coupling member, the lower coupling member, and the middlecoupling member and magnetic resonance tool of FIGS. 1 and 2;

FIGS. 8 and 9 are diagrammatic views of the upper coupling member, themiddle coupling member and magnetic resonance tool, and the lowercoupling member showing the current fields established in the radial andaxial modes of operation, respectively, of the embodiment of FIG. 1 inaccordance with the present invention;

FIGS. 10 and 11 are schematic diagrams of the equivalent circuits atzero phase angle unity power factor series resonance of the apparatus ofFIGS. 8 and 9, respectively;

FIG. 12 is a schematic diagram of the equivalent circuit of thephase-modulating power supply, cabling, upper coupling member, middlecoupling member and magnetic resonance tool and lower coupling member ofFIG. 1; and

FIGS. 13a-13 i are a series of curves showing the signals at variouspoints in the sideband detector of FIG. 1.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT

Referring to FIG. 1, a borehole 10 is shown in a formation 12 ofgeologic structures, the petrophysical properties of which are to beexamined using the apparatus and method of the present invention.Positioned within borehole 10 is a three-part resistivity well loggingassembly utilizing nuclear magnetic resonance. The assembly consists ofan upper coupling member 14, a lower coupling member 16, and a middlecoupling member and magnetic resonance tool 18 positioned therebetween.

The upper coupling member 14, the lower coupling member 16, and themiddle coupling member and magnetic resonance tool 18 are suspended inspaced relationship, as illustrated in FIG. 1, on a suitable cable 20.Cable 20 is routed above the borehole 10 over a pulley 22 which ismounted on a suitable support frame 24. The end of the cable 20 isattached to a suitable takeup spool (not shown) mounted on and driven bya motor 26.

The position of the cable 20 within borehole 10 is determined by aposition sensing roller 28 or other suitable depth metering device. Thecable depth information is fed to electrical supply and processingcircuitry 30 containing power source and control equipment and signalprocessing and recording equipment as will be more fully describedbelow. Supply and processing circuitry 30 is electrically connected tocable 20 by an electrical feed line 32.

In accordance with the present invention, the middle coupling member 18also contains a magnetic resonance tool which, when energized,establishes a magnetic resonance sensitive volume 34 in the generallycylindrical space surrounding the member 18 as diagrammaticallyillustrated in FIG. 1.

FIG. 2 is a partially pictorial and partially block diagram of thecomponents of the apparatus of FIG. 1 in accordance with the presentinvention. In FIG. 2, the radio frequency antenna of the magneticresonance tool of middle coupling member 18 is electrically coupled by aline 40 to a transmitter/receiver (T/R) matching circuit 42. The T/Rmatching circuit 42 typically includes a resonance capacitor, a T/Rswitch, and both transmit and receive matching circuits. The T/Rmatching circuit is coupled to an RF power amplifier 44 and to areceiver preamplifier 46.

System control is provided by a computer 48, which provides a controloutput to a pulse programmer 50. The pulse programmer 50 receives an RFinput from a variable frequency RF source 52, which is synchronized withthe phase modulating power supply 66 through cable 20, as illustrated.The pulse programmer 50 also controls the operation of an RF driver 54,which receives an input from variable frequency RF source 52 and outputsa signal to RF power amplifier 44.

The output of the RF receiver preamplifier 46 is supplied to an RFreceiver 56 which receives an input from a phase shifter 58 in responseto a signal received from the variable frequency RF source 52. Receiver56 is connected to a sideband detector 60 which, in accordance with thepresent invention, receives a reference signal from power supply 66through cable 20, demodulates and detects the output of receiver 56 andthen supplies the resulting output to an analogue to digital (A/D)converter 62. The output of the A/D converter 62 is fed to computer 48which, in turn, provides desired well logging output data to the supplyand processing circuitry 30 via a telemetry circuit 64.

Supply and processing circuitry 30, in accordance with the presentinvention, includes a phase-modulating power supply 66 which supplies“low” frequency electric power down the cable 20 to the coupling members14, 16 and 18, as will be more fully described hereinbelow, which alsocan be converted to supply power to the elements contained in block 70.Circuit 30 also includes a processor and recorder 68 which receives thedata from telemetry circuit 64 and position sensor 28, processes thesame, and records the information for further reference and analysis.

Many of the elements described above may be contained within a housing,diagrammatically shown as block 70, mounted in suitable fashion on,adjacent to or within the middle coupling member and magnetic resonancetool 18. Alternatively, some of the elements contained within block 70may be contained in separate housings or within the supply andprocessing circuitry 30 outside the borehole and above ground.

Each of the upper and lower coupling members 14 and 16, respectively,may be constructed in similar fashion in accordance with the presentinvention. An exemplary configuration for such members is illustrated inmore detail in FIGS. 3 and 4.

The coupling members may be constructed of a cylindrical steel tube 80having an outer radius r₂. Tube 80 is provided through its full lengthwith an axial hole 82 having a radius r₁. Hole 82 serves as a racewayfor cable 20. Wound around tube 80 is a primary winding 84, which has asecondary winding 86 wound concentrically thereover to form atransformer. The ends of tube 80 are provided with generally circularend plates 88 to contain the primary and secondary windings 84 and 86.An outer, generally cylindrical steel pipe 90, having an interior radiusr₃ and an exterior radius r₄, surrounds the entire assembly and ismechanically and electrically interconnected with the tube 80 and eachof the end plates 88 to form a self-contained unit.

The following dimensions are provided for each of the upper and lowercoupling members, assuming equal flux density in steel: r₁≅0.25 inches;r₂≅1.225 inches; r₃≅2.75 inches; r₄≅3 inches; and axial length≅7 feet.While the configuration described and illustrated in FIGS. 3 and 4 forthe upper and lower coupling members 14 and 16 is considered to be mostsuitable, any other desired configuration capable of providingcapacitive or conductive coupling to the interior of the borehole whileat the same time providing impedance matching, to bring the entirecircuit into zero phase angle unity power factor series resonance, maybe employed. To that end, and as will be described more fully below, thetransformer contained within each of the upper and lower couplingmembers 14 and 16, respectively, may be provided with an impedanceadjusting coil to assist in achieving zero phase angle unity powerfactor series resonance.

A preferred embodiment of sideband detector 60 is shown in FIG. 5. InFIG. 5, antenna 100 of the magnetic resonance tool 18 is coupled to animpedance matching circuit 102, feeding a noise matching preamplifier104 through a protection circuit 106. The output of the noise matchingpreamplifier is fed to an RF amplifier 108, the output of which isdirected to a double-balanced demodulator 110, 112. The output of thedouble-balanced demodulator 110, 112 is fed through a summing amplifier114 to an alternating current integrator 116 and an audio frequencyamplifier 118. Each of the demodulators 110 and 112 is supplied by aquadrature output from the magnetic resonance master radio frequencyoscillator. The demodulated outputs are then added at summer 114 toprovide a cross-correlated input to the integrator 116 (estimating thedirect current J_(o)( ) term) as well as to a low frequency amplifier118 that, in turn, feeds additional sets of double-balanced demodulators120, 121 each of whose added cross-correlated outputs are also summed bysummers 122 and integrated by integrators 124, thereby estimating therelative strength of each sideband element of the spectrum, J_(n)( ).

Turning to FIG. 6, the phase modulating power supply 66 of the presentinvention is illustrated in schematic form. The circuit shown in FIG. 6.includes upper and lower high power “low” frequency voltage sourcesV_(U) and V_(L), respectively, each capable of providing, for example,10 kilowatts of power at a frequency of 10 to 10,000 Hz. The uppervoltage source is connected to a polarity reversing switch 150 to enablethe output polarity to be selectively reversed depending upon whetheroperation in the radial or axial modes is desired, as will be describedmore fully below.

One side of each of the upper and lower voltage sources is coupledthrough a suitable current measuring device I_(N) to a neutral cableline 152. The other side of each of the respective upper and lowervoltage sources V_(U) and V_(L) is coupled through similar currentmeasuring devices I_(U) and I_(L) to upper and lower impedance adjustingcoils 154 and 156, as shown. The opposite ends of coils 154 and 156 areconnected to respective upper and lower cable lines 158 and 160. Thecable lines 152, 158 and 160 are electrically insulated from each otherand are bundled and fed down the borehole 10 with support cable 20 usingsuitable electrical cable feed techniques known in the art.

Turning now to FIG. 7, cable lines 152, 158 and 160 are fed down theborehole 10 with the upper cable line 158 connected through a suitablematching inductance to the primary coil of the transformer containedwithin the upper coupling member 14, as illustrated. The opposite end ofthe primary coil is coupled to one end of the secondary, the oppositeend of which is coupled to the steel housing described above andillustrated in FIGS. 3 and 4. The common connection of the primary andsecondary windings of the transformer 152 complete the circuit.

In similar manner, the lower cable 160 and the neutral cable 152 areconnected to a Faraday shield 162 which surrounds the magnetic resonancetool 18 and forms the middle capacitive coupling member in thetriple-coupling array.

FIGS. 8 and 9 illustrate the current field lines created in the boreholeand surrounding formation when the three coupling members 14, 16 and 18are energized in the radial and axial modes, respectively. The radialmode is presented when the polarity reversing switch 150 of the circuitillustrated in FIG. 6 is positioned to provide similar polarization ofthe upper and lower lines 158 and 160 relative to the neutral line 152.The current fields so created have a predominantly radial component withrespect to the sensitive volume of the magnetic resonance device. Theeffects of the axial current field along the borehole axis cancel (beingin opposite directions).

When the upper and lower coupling members are polarized in opposition, acurrent field is established with current field lines runningpredominantly parallel to the borehole axis as illustrated in FIG. 9.Only that component of the axially directed current field external tothe sensitive volume produces phase modulation and is measured. Theeffects of any radial components cancel (being equal but opposite aboveand below a plane transverse to, and containing the midpoint of, themagnetic resonance sensitive volume).

Equivalent circuits for the radial and axial modes of operation areillustrated schematically in FIGS. 10 and FIGS. 11, respectively, withthe resistance calculated and obtained in accordance with thestraightforward formulations set forth below.

In FIG. 12, a schematic diagram is illustrated showing the equivalentcircuit of the overall combination of the phase-modulating power supplycircuit 66; the cable feed lines 152, 158 and 160; and the down holetuning inductances and the transformers contained within the upper andlower coupling members, respectively. From the equivalent circuit, atzero phase angle unity power factor series resonance, accuratecalculations can be obtained of apparent resistance values in thegeologic formation.

FIGS. 13a-13 i illustrate the sidebands produced by the interaction ofthe phase-modulating electric power and the output of the magneticresonance tool. The figures also show the formulations used in thederivation of relevant data at different stages of the sidebanddetection process within the sideband detector 60.

The relationship between the magnetic resonance device depicted as atypical embodiment in applicant's U.S. Pat. No. 5,412,322, incorporatedherein by reference, and prior art well logging devices such as the MRILunit of the Numar Corporation, implies a conformal transformationbetween the orthogonal rectilinear coordinates, x′, y′, z′ and theorthogonal curvilinear cylindrical coordinates ρ, θ, z.

The Zeeman H_(o) field is oriented in the {overscore (l)}_(θ) directionin the well logging device and varies in strength radially. A pulsednarrow band radio frequency excitation field H₁ is applied perpendicularto the Zeeman H₀ field so as to excite only a narrow resonant cylinderof spins coaxial with the borehole. The radius and thickness of thiscylinder may be changed by altering the H₁ field frequency content.

A gradient field {overscore (G)} is defined by applicant's U.S. Pat. No.5,412,322:${\rho \overset{\_}{G}\quad \underset{\underset{\_}{\_}}{\Delta}{\nabla\quad \left( {\rho \quad h_{\theta}} \right)}} = {\overset{\_}{J}x\quad \rho {\overset{\_}{1}}_{\theta}}$$\overset{\_}{J} \equiv {{{\overset{\_}{1}}_{\rho}i_{\rho}} + {{\overset{\_}{1}}_{\theta}i_{\theta}} + {\overset{\_}{k}i_{z}}}$

where {overscore (J)} is the applied phase modulating current density,assuming circular symmetry in a cylindrical system. (Appendix 1).

A conventional alternating phase Carr-Purcell-Meibom-Gill (CPMG)sequence is utilized to excite (π/2 pulse) and rephase (π pulse) thespins in the sensitive volume. The interval between π pulses 2Tcp can bemade short enough (e.g. 1.5-5 ms) to both permit the estimation offormation pore size distribution, by regression analysis of the echotrain, and to exclude signals from surface associated fluid, such as inshaly sands.

The echo train is composed of a sum of exponential decays whose timeconstants are strongly dependent on the surface to volume ratio of eachpore size population. Regression analysis then can estimate the relativemagnitude of each population of pore sizes permitting separation of thetotal fluid volume (porosity) into surface associated (bound volumeirreducible BVIR) fluid and bulk fluid. Signals associated with fluid inshaly sands are excluded by the initial short Tcp pulse interval, duringwhich time passive diffusion brings surface associated spins close toboundaries which permit spin-lattice relaxation T₁ and rapid T₂dephasing, eliminating signals from such spins at a much greater ratethan by the dephasing caused by passive diffusion across weakergradients in bulk fluid. See, for example, U.S. Pat. No. 5,539,309 toVan Wyk; U.S. Pat. No. 5,557,200 to Coates; and U.S. Pat. No. 5,565,775to Stallmach; and articles by Slichter, C. P. “Principles of MagneticResonance”, 3rd Edition, 1989, Appendix G; and Stejskal, E. O., J. Chem.Phys., Vol. 43, number 10, 15 November 1965, p. 3597-3603.

For the purpose of this invention, the Tcp interval may be prolongedduring all of, or the later part of, the pulse sequence from the 1.5-5ms, as is used for estimating surface associated fluid volume toestimate bulk fluid volume. The later portion of the echo train thenconsists of signals from spins associated with bulk fluid, relativelyfree of surface or diffusion effects, which fluid is composed of anaqueous phase and may also have a hydrocarbon phase.

The spins in bulk fluid are less affected by wall induced susceptibilityvariations and spin-lattice relaxation and therefore have a narrowerbandwidth, producing a more prolonged echo signal, permitting a longer“read” time. This longer signal is preferred for detection of the phasemodulation of the spins produced by the time varying gradient created bythe time-varying formation current whose frequency is much less than theLarmor frequency, but whose period is much less than the “read” time ofthe echo. A phase encoding gradient of 10 KHz is compatible with a“carrier” Larmor frequency of 700-1200 KHz and an intrinsic gradient of17 gauss per cm. (U.S. Pat. No. 5,696,448 to Coates.)

Both the estimation of surface associated fluid volume (BVIRR) and themeasurement of formation current by detecting the resulting phasemodulation of the spins in bulk fluid can be accomplished with a singlepulse sequence consisting of an initial short Tcp echo train of 1.5-5 msfollowed by several longer Tcp intervals.

The interval between pulse sequences T_(r) (or W wait time) is made longenough to allow relaxation of the hydrocarbon of interest (U.S. Pat. No.5,497,087 and U.S. Pat. No. 5,498,960, both to Vinegar) but short enoughto permit rapid data acquisition (U.S. Pat. No. 5,309,098 to Coates; andU.S. Pat. No. 5,486,762 to Freedman).

This invention adds two systems of electrodes in the borehole topre-existing magnetic resonance well logging devices:

A conductor array (such as a Faraday shield) with electrodesperpendicular to the E₁ component of the RF field, arranged coaxial tothe borehole and placed around the magnetic resonance device so as notto affect the RF field, creates a current field with a predominatelyradial component with respect to the sensitive volume of the magneticresonance device when indifferent electrodes are symmetrically placedwithin the borehole above and below the magnetic resonance device. Onlythe radial component of the current field directly perpendicular to theZeeman H₀ field is measured by its effect in producing phase modulationof the magnetic resonance signal. This radial component is measured andseparated from the total current flow. The effects of the axial currentfield along the borehole axis cancel (being in opposite directions).This permits an estimation of the apparent resistance of the boreholeprofile and of the formation near the sensitive magnetic resonancevolume, in a series distribution. (Appendix 3).

Electrodes symmetrically placed within the borehole above and below themagnetic resonance device and oppositely polarized produce a currentfield predominately parallel to the borehole axis. Only that componentof this axially directed current field external to the sensitive volumeproduces phase modulation and is measured. This permits separateestimates of the resistance caused by borehole effects within themagnetic resonance sensitive volume and the desired resistance of theformation external to the magnetic resonance sensitive volume. Selfinduction tends to distribute the axial current flow peripherally (“skineffect”), decreasing the participation of the invaded zone in thislatter measurement.

Changes in phase modulation produced by altering these current flows canbe used to provide estimates of spin-spin relaxation time T₂ orBloch-Torrey bulk diffusion coefficient D (U.S. Pat. No. 5,212,447 toPaltiel; U.S. Pat. No. 5,565,775 to Stallmach) which correlate withvarious petrophysical parameters. (Appendix 4).

The total current {overscore (J)} consists of i_(z) parallel to theborehole and i_(ρ) radial to the borehole, since i_(θ)=0. By theMaxwell-Ampere equation (in cylindrical coordinates):

{overscore (V)}X{overscore (H)}={overscore (J)}

Since$\frac{\partial h_{\rho}}{\partial\theta} \equiv \frac{\partial h_{\theta}}{\partial\theta} \equiv \frac{\partial h_{z}}{\partial\theta} \equiv 0$

by cylindrical symmetry, $\begin{matrix}{i_{\rho} = {{{- \frac{1}{\rho}}\frac{\partial\left( {\rho \quad h_{\theta}} \right)}{\partial z}\underset{\underset{\_}{\_}}{\Delta}} - G_{z}}} & \left( {{Appendix}\quad 1} \right)\end{matrix}$

then, $\begin{matrix}{i_{z} = {{\frac{1}{\rho}\frac{\partial\left( {\rho \quad h_{\theta}} \right)}{\partial\rho}\underset{\underset{\_}{\_}}{\Delta}} - G_{\rho}}} & \left( {{{Appendix}\quad 1},2} \right) \\{and} & \quad \\{0 = {\frac{\partial h_{\rho}}{\partial z} - \frac{\partial h_{z}}{\partial\rho}}} & \quad\end{matrix}$

Any h_(ρ) or h_(z) is inconsequential since {overscore(l)}_(ρ)h_(ρ)+{overscore (l)}_(θ)H₀≅{overscore (l)}_(θ)H₀ andkh_(z)+{overscore (l)}_(θ)H₀≅{overscore (l)}_(θ)H₀ to first order.(Slichter, C. P., Principles of Magnetic Resonance, Third edition,Springer-Verlag, Berlin; 1989, eq 7.376, 7.377, p. 358).

If the frequency of the phase modulating current is a multiple of theCarr-Purcell interval, then the bulk diffusion D can be separated fromthe spin-spin relaxation T₂ by regression analysis. Further, when thephase modulation frequency Ω is much greater than the gradient strengthG there is little loss of signal resulting from diffusion through theoscillating gradient. (Appendix 4). (U.S. Pat. No. 5,565,775 toStallmach).

In the radial connection, the current component i_(ρ), perpendicular tothe magnetic resonance sensitive volume, produces a gradient {overscore(G)} of the Zeeman Field H₀ in the Z-direction. Periodically varyingi_(ρ) by applying a periodic voltage between the central coaxialelectrode array placed around the magnetic resonance device in theborehole and the symmetrically placed electrodes placed in the boreholeabove and below the magnetic resonance device phase modulates themagnetic resonance echo. This is an effect dependent only on thefrequency and strength of i_(ρ) at, and directly perpendicular to, thesensitive volume of the magnetic resonance device as described morefully in applicant's U.S. Pat. No. 5,412,322 (Appendix 3). The measuredradial current density, i_(ρ), is perpendicular to the sensitive volumeof the magnetic resonance device. This conduction path is stronglyinfluenced by borehole and invaded zone resistivity, as distinguishedfrom the more peripheral formation resistivity.

In the axial connection, any closed circumferential line integral withinthe cylindrical sensitive volume, but drawn about the borehole axis,defines a set of areas each of which includes the current flowing inboth the borehole itself and the formation within the sensitive volume,as well as the current in the supply wire to the down-hole electrode.This adds to zero, in the quasi-static state, if no current flowsoutside of the sensitive volume. If however, current flows in theformation surrounding, and peripheral to, the magnetic resonancesensitive volume, this peripheral current then represents the differencebetween the current in the supply wire and the total current within thesensitive magnetic resonance volume, and, by Stokes Law, produces amagnetic field h_(θ) which can phase modulate the spins in the sensitivevolume, separating the two current components internal and external tothe sensitive volume. (Appendices 2 and 5).

Self-inductance in this longer axially distributed current path produceseddy currents resulting in a greater component of flow of current in themore peripheral formation, particularly at higher frequencies and higherconductivity. (Smythe, W. R., “Static and Dynamic Electricity”,McGraw-Hill Book Company, New York 1950, Chapter XI).

The two approximate equivalent circuits for the radial and axial modesof operation, at zero phase angle unity power factor series resonance,are shown in FIGS. 10 and 11, respectively, and are analyzed as follows:

a) In the radial case (FIG. 10):

^(r) a _(ρ)=(r _(m) +r _(mc) +r _(xo))+(½)(r _(t))+(½)(r _(m) +r _(mc)+r _(xo))

^(r) a _(ρ)=(1½)r _(b)+(½)r _(t)

b) In the axial case (FIG. 11):

^(r) a _(k)=(r _(m) +r _(mc) +r _(xo))+2r _(t)+(r _(m) +r _(mc) +r_(xo))

 ^(r) a _(k)=2r _(b)+2r _(t)

 where b, m, mc, xo, t and a pertain to borehole, mud, mudcake, invadedzone, formation, and apparent resistance, respectively.

Each resistance r is the ratio of the periodic voltage applied to theconducting formation and the axial, i_(z), or radial, i_(ρ), current asmeasured by detection of the amplitude of the sidebands in the magneticresonance signal.

Solving for r_(t:)

r _(t)=(¾)r _(ak) −r _(aρ)

which is free of borehole and invaded zone effects.

For a non-conducting borehole, cylindrical electrodes of tool diameterof, for example, 6 inches, are preferably centered and stabilized in theborehole by a suitable structural component, such as a tool pad, a skid,an array of metal “bowstrings,” or the like, which serve to increase thecapacitive coupling, to center the tool in the borehole (reducing toolmotion dephasing), and can provide electrical contact with thesurrounding formation. The capacitance between a seven-foot long,six-inch diameter tool and any surrounding conducting formation near thesensitive volume can be expected to be very small, e.g. 20-200 micromicro Farads, assuming:$C = {\frac{K}{18{\ln \left( {r_{2}/r_{1}} \right)}} \times 10^{- 9}{F/m}}$

for a cylindrical capacitor, where K is the dielectric constant (about 2for oil), r₁ is the radius of the electrode, and r₂ is the effectiveradius of the surrounding conducting formation. The capacitive reactancefor oil at 1-10 KHz can be expected to be high, on the order of, forexample, 0.008-6 megohms per electrode.

If the electrodes are connected to the secondary winding of an impedancematching transformer having a secondary to primary turns ratio “a,” thesecondary impedance will be reflected into the primary circuit dividedby that ratio “a²,” reducing its magnitude. A primary loading inductancecan then bring the entire circuit into zero phase angle unity powerfactor series resonance. This measures and cancels out the unknowncapacitive reactances, leaving only the known equivalent circuitresistances of the wire line and transformer in series with the unknowntotal apparent formation resistance r_(a).

The impedance matching transformer and most or all of the loadinginductance can be made integral with the tool in the borehole, improvingthe power transmission efficiency of the wire line. The apparatus can bemade partially tunable by a variable impedance at the wellhead to obtainzero phase angle unity power factor series resonance under varying toolenvironments.

Using two impedance matching circuits employing either a transformer oran autotransformer with variable impedances and separate metering ineach current loop permits separate adjustment of each current loop tozero phase angle unity power factor series resonance. When the supplyvoltages and currents are equal, at series resonance, the tool iscentered in the bed (U.S. Pat. No. 5,550,473 to Klein). The appliedvoltage less the known IR voltage drop through the known losses in thetool is the voltage applied to any conducting formation face opposingthe tool electrodes.

The total current through the electrode array is measured at thewellhead. Phase modulation of the spins in the sensitive volume of themagnetic resonance device allows separation of this total current intotwo components; either into one component flowing axially through theborehole environment within the sensitive volume of the magneticresonance device and another component flowing axially through theformation surrounding this sensitive volume (in the axial connection),or into one radial component perpendicular to the sensitive volume ofthe magnetic resonance device and another component flowing along, andin the proximity of, the borehole axis (in the radial connection).

Empirical determination of geometric factors permits estimation offormation resistivity R_(t) peripheral to the tool and invaded zoneresistivity R_(xo) (if any) near the tool from the measured resistancer_(a), in each case. See, Bassiouni, op. cit. p. 107-111.

The following steps will yield a measurement of total current andapplied voltage at the conducting formation surface:

a) In the radial case:

1. Select {circumflex over (V)}_(u) and {circumflex over (V)}_(L)polarity for radial connection.

2. Select {circumflex over (Z)}_(u) and {circumflex over (Z)}_(L) forzero phase angle unity power factor.

3. Select {circumflex over (V)}_(u) and {circumflex over (V)}_(L) sothat${{\hat{V}}_{L} - {\hat{V}}_{U}} = \left( {{\hat{I}}_{U} + {{\hat{I}}_{L}\left( {R_{W} + R_{\rho} + \frac{R_{s}}{a^{2}}} \right)}} \right.$

then:${\hat{V}}_{\rho} = {{a\left( {{\hat{V}}_{L} - {{\hat{I}}_{L}\left( {{2R_{W}} + R_{\rho} + \frac{R_{s}}{a^{2}}} \right)} + {{\hat{I}}_{U}R_{W}}} \right)}\quad {or}}$${\hat{V}}_{\rho} = {a\left( {{\hat{V}}_{U} + {{\hat{I}}_{U}\left( {{2R_{W}} + R_{\rho} + \frac{R_{s}}{a^{2}}} \right)} - {{\hat{I}}_{L}R_{W}}} \right)}$

with Î_(ρ) determined by the amplitude of the second sideband of themagnetic resonance signal (Appendix 3):

|V _(ρ) /I _(ρ)|≅1½r _(b)+½r _(t) Δr _(aρ)

b) In the axial case:

1. Adjust {circumflex over (Z)}′_(u)={circumflex over (Z)}′_(L) forborehole environment to set tuning range for {circumflex over (Z)}_(u)and {circumflex over (Z)}_(L).

2. Select {circumflex over (V)}_(u) and {circumflex over (V)}_(L)polarity for axial connection.

3. Adjust {circumflex over (V)}_(u) and {circumflex over (V)}_(L) forÎ_(u)=Î_(L) and Î_(N)=0.

4. Tune with {circumflex over (Z)}_(u) and {circumflex over (Z)}_(L) formaximum Î_(u) and Î_(L) with zero phase angle and unity power factor.

5. Repeat 3 and 4 as needed.

 then, with Î_(u)=Î_(L) ΔÎ:${{\hat{V}}_{u} + {\hat{V}}_{L}} = {\hat{I}\left( {{2R_{w}} + {2R_{\rho}} + \frac{2R_{s}}{a^{2}} + \frac{r_{a}}{a^{2}}} \right)}$

the voltage applied at the conducting formation face is:${\hat{V}}_{t} = {a\left\lbrack {\left( {{\hat{V}}_{u} + {\hat{V}}_{L}} \right) - {2{\hat{I}\left( {R_{w} + R_{p} + \frac{R_{s}}{a^{2}}} \right)}}} \right\rbrack}$

With I_(t) determined by the amplitude of the first sideband of themagnetic resonance signal—(Appendix 3):${\frac{V_{t}}{I_{t}}} = {{r_{bu} + r_{tu} + r_{tl} + r_{bl}} \cong {{2r_{b}} + {2r_{t}\underset{\underset{\_}{\_}}{\Delta}r_{ak}}}}$

The apparatus described herein permits estimation of formationresistivity R_(t) and invaded zone resistivity R_(xo) usingexperimentally determined empirical geometric factors relating measuredresistance to resistivity.

(The early short Tcp component of the CPMG sequence can be used toestimate the surface associated irreducible water saturation, free ofclay effects, as the bound volume irreducible component (BVIRR) of totalporosity (U.S. Pat. No. 5,557,200 to Coates).

The component of total porosity free of solid-liquid interface surfaceeffects can be estimated from the amplitude of the magnetic resonancesignal during the later long Tcp portion of the CPMG sequence. Thiscomponent of total porosity contains spins from water and may containspins from producible non-conducting hydrocarbons (U.S. Pat. No.5,539,309 to Van Wyk).

The total porosity, excluding the water associated with high surfacearea shales or clays, is termed the effective porosity and can beestimated by regression analysis of a complete short Tcp CPMG echo train(1-3 ms) (U.S. Pat. No. 5,557,200 to Coates).

The relaxation time T_(r) (or wait time W) may be adjusted to create T₁saturation, or allow T₁ relaxation, of spins from natural gas, creatingmultiple selective CPMG echo trains that can be subtracted to estimatethe presence of and a restricted diffusion coefficient of the gas (U.S.Pat. Nos. 5,497,087 and 5,498,960, both to Vinegar).

The amplitude or frequency of the applied phase-modulating current fluxcan be varied to permit estimation of the unrestricted bulk diffusioncoefficient D or the intrinsic spin-spin relaxation T₂ of the bulk fluidthat is free of surface effects (U.S. Pat. No. 5,565,775 to Stallmach).(Appendix 4).

The parameters of the sequences can be continuously altered to providethe optimum statistical reliability with the minimum logging time (U.S.Pat. No. 5,309,098 to Coates; U.S. Pat. No. 5,486,762 to Freedman; andU.S. Pat. No. 5,517,115 to Prammer).

The continuous efficient logging of R_(t) and R_(xo), together with thecontinuous measurement of the various components of porosity, therestricted and unrestricted diffusion coefficients, and the various T₁and T₂ relaxation coefficients, permits graphical computation of theformation water resistance R_(w′) (Theory, Measurement andInterpretation of Well Logs; Bassiouni, Z.; SPE Textbook series, Volume14, Richardson, Tex., 1994, p. 276); the single m−n exponent w forirreducible water saturation (W₁) and for water filled zones (W_(W))(U.S. Pat. No. 5,412,320 to Coates); and of log R_(xo)/R_(t) quick-lookreconnaissance for zones with movable hydrocarbons (Bassiouni; op. cit.,p. 252).

It should be clear to those skilled in the art that the method of thepresent invention can also be used in measuring properties of samples ofmaterials in a laboratory setting, as well as in situ logging-typeincluding logging/measuring while drilling (LWD/MWD) measurements.

Numerous other combinations and permutations of functions of the dataproduced by the well logging tool will be apparent to the experiencedwell-log analyst, petrogeologist, and geophysicist.

Inasmuch as the present invention is subject to many variations,modifications and changes in detail it is intended that all mattercontained in the foregoing description, the attached appendices or theaccompanying drawings shall be interpreted as illustrative and not in alimiting sense.

APPENDIX 1 Mathematical Relationship Between Applicant's U.S. Pat. No.5,412,322 and The Present Invention

Consider an orthogonal {overscore (i)}, {overscore (j)}, {overscore (k)}rectilinear space x′, y′, z′ bounded by

0≦x′; y′≦2r; −∞<z′<+∞; with {overscore (i)}×{overscore (j)}={overscore(k)}

Transform this space to an orthogonal cylindrical system with circularsymmetry θ, ρ, Z such that

dz′=ρdθ{overscore (i)}×{overscore (j)}={overscore (k)}

dx′=dρ{overscore (1)}_(ρ) ×{overscore (k)}={overscore (1)}_(θ)

dy′=dz

The gradient {overscore (G)}′Δ{overscore (∇)}h_(z′). The Larmorfrequency ω=

γ(H0+∫{overscore (G)}′·d{overscore (s)}′); d{overscore (s)}′Δ={overscore(i)}dx′+{overscore (j)}dy′+{overscore (k)}dz′

ρ{overscore (G)} Δ{overscore (∇)}(ρh _(θ)); ω=γ(H0+∫{overscore(G)}·d{overscore (s)}); d{overscore (s)}={overscore (1)}_(ρ)dρ+{overscore (k)}dz+{overscore (1)}_(θ) dθ

$i_{z^{\prime}} \equiv \frac{\partial h_{x^{\prime}}}{\partial z^{\prime}} \equiv \frac{\partial h_{y^{\prime}}}{\partial z^{\prime}} \equiv \frac{\partial h_{z^{\prime}}}{\partial z^{\prime}} \equiv 0 \equiv \frac{\partial h_{\theta}}{\partial h_{\theta}} \equiv \frac{\partial h_{\rho}}{\partial\theta} \equiv \frac{\partial h_{z}}{\partial\theta} \equiv i_{\theta}$

 {overscore (J)}′Δ{overscore (i)}i _(x′) +{overscore (j)}i _(y′)+{overscore (k)}i _(z′) ; {overscore (J)} Δ{overscore (1)}_(ρi) _(ρ)+{overscore (k)}i _(z)+{overscore (1)}_(θ) i _(θ)

{overscore (H)}′Δ{overscore (i)}h _(x′) +{overscore (j)}h _(y′)+{overscore (k)}(H0+h _(z′)); {overscore (H)} Δ{overscore (1)}_(ρ) h_(ρ) +{overscore (K)}h _(z)+{overscore (1)}_(θ)(H0+h _(θ))

${\overset{\_}{\nabla}{\times \quad \overset{\_}{H^{\prime}}}} = {{{\overset{\_}{i}\frac{\partial h_{z^{\prime}}}{\partial y^{\prime}}} + {\overset{\_}{j}\frac{\partial h_{z^{\prime}}}{\partial x^{\prime}}} + {\overset{\_}{k}(0)}} = {\overset{\_}{J^{\prime}}\quad \text{(Maxwell-Ampere)}}}$${\overset{\_}{k} \times \left( {\overset{\_}{\nabla}{\times {\overset{\_}{H}}^{\prime}}} \right)} = {{{\overset{\_}{i}\left( \frac{\partial h_{z^{\prime}}}{\partial x^{\prime}} \right)} + {\overset{\_}{j}\left( \frac{\partial h_{z^{\prime}}}{\partial y^{\prime}} \right)}} \equiv {{\overset{\_}{\nabla}h_{z^{\prime}}}\underset{\underset{\_}{\_}}{\Delta}{\overset{\_}{G}}^{\prime}}}$

 therefore, {overscore (K)}×{overscore (J)}′={overscore (G)}′; |J|=|G′|.q.e.d.

${\overset{\_}{\nabla}{\times \quad \overset{\_}{H}}} = {{{{- {\overset{\_}{1}}_{\rho}}\frac{1}{\rho}\frac{\partial\left( {\rho \quad h_{\theta}} \right)}{\partial z}} + {{\overset{\_}{1}}_{\theta}(0)} + {\overset{\_}{k}\frac{1}{\rho}\frac{\partial\left( {\rho \quad h_{\theta}} \right)}{\partial\rho}}} = {{{\overset{\_}{J}\left( {\overset{\_}{\nabla}{\times \quad \overset{\_}{H}}} \right)} \times \quad \rho \quad {\overset{\_}{1}}_{\theta}} = {{{\overset{\_}{k}\frac{\partial\left( {\rho \quad h_{\theta}} \right)}{\partial z}} + {{\overset{\_}{1}}_{\rho}\frac{\partial\left( {\rho \quad h_{\theta}} \right)}{\partial\rho}}} \equiv {{\overset{\_}{\nabla}\left( {\rho \quad h_{\theta}} \right)}\underset{\underset{\_}{\_}}{\Delta}\rho \quad \overset{\_}{G}}}}}$

 therefore, {overscore (J)}×ρ{overscore (1)}_(θ) =ρ{overscore (G)};|J|=G| q.e.d.

APPENDIX 2 Magnetic Field in the Axial Mode

${i_{z} = {\frac{1}{\rho}\frac{\partial\left( {\rho \quad h_{\theta}} \right)}{\partial\rho}}};{{i_{z}\rho {\rho}} = {\left( {\rho \quad h_{\theta}} \right)}}$∫₀^(r)i_(z)2πρρ = 2π∫₀^(rh_(θ))(ρ  h_(θ)) I = 2π  rh_(θ)$h_{\theta} = \frac{I}{2\pi \quad r}$

which follows more directly from Stokes' Law.

APPENDIX 3

Frequency Spectrum and Relative Strength of Magnetic Resonance Signal

In SI units, ω=γH with H in amperes/meter and γ in meters/ampere-second.The Zeeman field H₀ creating precession about the {overscore (1)}_(θ)axis consists of a static portion H_(θ) created by the magneticresonance tool permanent magnets and a superimposed oscillating fieldh_(θ) of frequency Ω created by the current flux J perpendicular to theZeeman field H_(o) (Wollin, U.S. Pat. No. 5,412,322).

Then: $\begin{matrix}{H_{O} = {H_{\theta} + {h_{\theta}\cos \quad \Omega \quad t}}} & \left( {{Appendix}\quad 1} \right) \\{{\overset{\_}{J} \times \overset{\_}{H}\frac{o}{Ho}} = {\overset{\_}{G} = {\frac{1}{\rho}{\nabla\rho}\quad h_{\theta}}}} & \quad\end{matrix}$

a) In the radial connection:${\quad h_{\theta}} = {{G_{z}{z}} = {{\frac{1}{\rho}\frac{\partial\left( {\rho \quad h_{\theta}} \right)}{\partial z}{z}} = {{- i_{\rho}}{z}}}}$h_(θ) = −i_(ρ)z 2π  rLi_(ρ) = I_(ρ)$h_{\theta} = {{- \frac{I_{\rho}}{2\pi \quad {rL}}}z}$

b) In the axial connection:${h_{\theta}} = {{G_{\rho}{\rho}} = {{\left( {\frac{1}{\rho}\frac{\partial\left( {\rho \quad h_{\theta}} \right)}{\partial\rho}} \right){\rho}} = {i_{z}{\rho}}}}$

for a constant radius cylinder r $\begin{matrix}{h_{\theta} = \frac{I_{t}}{2\pi \quad r}} & \left( {{{Appendix}\quad 2},5} \right)\end{matrix}$

I_(t) being the total axial current flowing within the cylinderboundary.

Since

ω=γH,

then

ω₀ =γH ₀ +γh _(θ) cos ωt.

The phase acquired by each spin during time t is$\Phi_{o} = {{\int_{o}^{t}{\omega_{o}{t}}} = {{\gamma \quad H_{\theta}t} + {\frac{\gamma \quad h_{\theta}}{\Omega}\sin \quad \Omega \quad t}}}$${{\Phi_{o}\underset{\underset{\_}{\_}}{\Delta}\omega_{\theta}t} + {u\quad \sin \quad \Omega \quad t}};{u\quad \underset{\underset{\_}{\_}}{\Delta}\frac{\gamma \quad h_{\theta}}{\Omega}}$

In the sensitive volume of constant spin density containing a uniformmagnetization of m spins per unit length, the total magnetization is:M̂ = ∫_(−L/2)^(+L/2)m  ^(j  Φ)z

Taking the Fourier transform with respect to time:  M̂ = ∫_(−∞)^(+∞)M̂^(jω  t)t = ∫_( = L/2)^(+L/2)m∫_(−∞)^(+∞)^(j(Φ − ω  t))tz

Defining${p = \frac{\omega_{o} - \omega}{\Omega}};{u = \frac{\gamma \quad h_{\theta}}{\Omega}};{\varphi = {\Omega \quad t}}$

yields${\quad \hat{M}} = {\int_{{- L}/2}^{{+ L}/2}{{m\left( \frac{1}{\Omega} \right)}{\oint_{complex}{^{j{({{p\quad \varphi} - {u\quad \sin \quad \varphi}})}}{\varphi}{z}}}}}$

by replacing φ with a complex variable and integrating in the complexplane. Then:${\quad \hat{M}} = {\left( \frac{\pi}{\Omega} \right){\int_{{- L}/2}^{{+ L}/2}{{{mJ}_{n}(u)}{z}}}}$

for integral values of n. (Sommerfeld: Math. Ann. 47,335,1896).

a) In the axial connection: M = mL$u = {\frac{\gamma}{\Omega} \cdot \frac{I_{t}}{2\pi \quad r}}$${\quad \hat{M}} = {M\quad \frac{\pi}{\Omega}{J_{n}\left( {\frac{\gamma}{2\pi \quad r\quad \Omega}I_{t}} \right)}}$

where M is the total number of spins in the sensitive volume.

b) In the radial connection:${u = {\frac{- \gamma}{\Omega}i_{\rho}z}};{{2\pi \quad {rLi}_{\rho}} = I_{\rho}};{u = {{- \frac{\gamma}{\Omega}}\frac{I_{\rho}}{2\pi \quad {rL}}z}}$${\quad \hat{M}} = {M \cdot \frac{\pi}{\Omega} \cdot \frac{1}{L} \cdot {\int_{{- L}/2}^{L/2}{{J_{n}\left( {\frac{{- \gamma}\quad I_{\rho}}{2\pi \quad {rL}\quad \Omega}z} \right)}{z}}}}$

for n even only, since J_(n)(u)=−J_(n)(−u) for n odd

Approximations

For small values of u:${{J_{n}(u)} \cong \frac{u^{n}}{n^{\prime}2^{n}}},{and}$${{J_{o}(u)} \cong 1};{{J_{1}(u)} \cong {\frac{1}{2}u}};{{J_{2}(u)} \cong {\frac{1}{8}u^{2}}}$

a) In the axial connection:

For the first sideband J₁(u)${\quad \hat{M}} = {\hat{M}\frac{\gamma}{4r\quad \Omega^{2}}I_{t}}$

b) In the radial connection:

For the second sideband J₂(u) $\begin{matrix}{{\quad \hat{M}} = {M \cdot \frac{\pi}{\Omega \quad L} \cdot \frac{1}{8} \cdot \left( \frac{{- \gamma}\quad I_{\rho}}{2\pi \quad {rL}\quad \Omega} \right)^{2} \cdot \frac{L^{3}}{12}}} \\{= {M \cdot \frac{1}{384\pi \quad r^{2}} \cdot \frac{\gamma^{2}}{\Omega^{3}} \cdot I_{\rho}^{2}}}\end{matrix}$

γ≅34π² meters/amp-seconds for protons.

The strength of the sidebands J₁(u) and J₂(u) relative to the centralLarmor frequency J_(o)(u), for small values of the argument u, is:

a) In the axial connection:${\quad \frac{{\hat{M}}_{1}}{{\hat{M}}_{o}}} = {{\frac{1}{2}u} = {{\frac{1}{4\pi \quad r} \cdot \frac{\gamma \quad I_{t}}{\Omega}} \cong {\frac{1}{4{\pi ({.2})}} \cdot \frac{34\pi^{2}I_{t}}{2\pi \quad f_{\rho}}} \cong {21.2\frac{I_{t}}{f_{p}}}}}$

for${f_{p} = {10^{4}{Hz}}},{\frac{_{1}}{_{o}} = {21.2 \times 10^{- 4}I_{t}}}$

b) In the radial connection: $\begin{matrix}{{\quad \frac{{\hat{M}}_{2}}{{\hat{M}}_{o}}} = {\frac{\frac{1}{8}{\left( \frac{{- \gamma}\quad I_{\rho}}{2\pi \quad {rL}\quad \Omega} \right)^{2} \cdot \frac{L^{3}}{12}}}{L} = {\frac{1}{8\left( {2\pi \quad r} \right)^{2}(12)} \cdot \left( \frac{\gamma \quad I_{\rho}}{\Omega} \right)^{2}}}} \\{= {{\frac{1}{15.4\pi^{2}} \cdot \left( \frac{\gamma \quad I_{\rho}}{\Omega} \right)^{2}} = {\frac{1}{15.4\pi^{2}} \cdot \left( \frac{34\pi^{2}}{2\pi} \right) \cdot \left( \frac{I_{\rho}}{f_{p}} \right)^{2}}}} \\{= {{\frac{(17)^{2}}{15.4} \cdot \left( \frac{I_{\rho}}{f_{p}} \right)^{2}} \cong {18.8\left( \frac{I_{\rho}}{f_{p}} \right)^{2}}}}\end{matrix}$

for${f_{p} = {10^{4}{Hz}}},{\frac{_{2}}{_{o}} = {188 \times 10^{- 9}I_{\rho}^{2}}}$

APPENDIX 4 Bloch-Torrey Bulk Diffusion—(Periodic Gradient)

$\frac{\hat{M}}{t} = {{- \frac{\hat{M}}{T_{2}}} - {j\hat{M}\gamma \quad {Gz}\quad \sin \quad \Omega \quad t} + {D{\nabla^{2}\hat{M}}}}$

Let$\hat{M} = {M_{o}^{{- t}/T_{2}}^{\frac{{+ {j\gamma}}\quad {Gz}\quad \cos \quad \Omega \quad t}{\Omega}}A_{t}}$

${\frac{1}{A}\frac{\delta \quad A}{\delta \quad t}} = {\left( {D\left( {j\frac{\gamma \quad G}{\Omega}\cos \quad \Omega \quad t} \right)} \right)^{2} = {{- \left( {D\left( \frac{\gamma \quad G}{\Omega} \right)} \right)^{2}}\cos^{2}\Omega \quad t}}$${\ln \quad A} = {{{- \left( {D\left( \frac{\gamma \quad G}{\Omega} \right)} \right)^{2}}{\int{\cos^{2}\Omega \quad t{t}}}} = {{- \left( {D\left( \frac{\gamma \quad G}{\Omega} \right)} \right)^{2}}\left( {\frac{t}{2} + \frac{\sin \quad 2\quad \Omega \quad t}{4\Omega}} \right)}}$

Burington, R. S.; Handbook of Mathematical Tables and Formulas; HandbookPublishers, Inc. Sandusky, Ohio, 1949.$\hat{M} = {M_{o}{^{j\quad \frac{\gamma \quad G}{\Omega}z\quad \cos \quad \Omega \quad t}\left\lbrack {^{- \frac{t}{T_{2}}}^{{- {({D{(\frac{\gamma \quad G}{\Omega})}})}^{2}}{({{t/2} + \frac{\sin \quad 2\quad \Omega \quad t}{4\quad \Omega}})}}} \right\rbrack}}$

when t=2τ the phase terms e^(j+) and e^(j−) cancel.

If ${\Omega = {n\frac{\pi}{4\tau}}},$

for n odd,

then sin 2Ω(2τ)=0 and cos Ω(2τ)=0.

Then$\hat{M} = {{M_{o}^{{- 2}{\tau {\lbrack{{1/T_{2}} + {\frac{1}{2}{D{(\frac{\gamma \quad G}{\Omega})}}^{2}}}\rbrack}}}\quad {or}\quad \ln \quad \hat{M}} = {{\ln \quad M_{o}} - {2{\tau \left( \frac{1}{T_{2}} \right)}} - {\left\lbrack \left( {\tau \left( \frac{\gamma \quad G}{\Omega} \right)} \right)^{2} \right\rbrack (D)}}}$

After Slichter, C. P. “Principles of Magnetic Resonance”Springer-Verlag, 3rd Edition, 1989, Appendix G, γ and M₀ are constantsin each experiment. G, Ω and τ are operator adjustable. D and T₂ are tobe measured. |G|=|J|; τ≡τ Carr-Purcell; Ω=2 πf_(p) (f_(p) is thefrequency of J).

APPENDIX 5 Analysis of the Axial Circuit

Consider a closed line integral in the 1 _(θ) direction through thecylindrical circular sensitive volume of radius ρ of the magneticresonance logging tool; wherein the Zeeman Field is of magnitudeH₀=H_(θ)+h_(θ) with ω₀=γH_(θ) being the central frequency of the narrowband H₁ radio frequency excitation field. Then, in the quasi-staticstate, Stokes Law yields${\oint{{\overset{\_}{H} \cdot d}\overset{\_}{s}}} = {{\int{\int{{\left( {\overset{\_}{\Delta} \times \overset{\_}{H}} \right) \cdot d}{\overset{\_}{a}.\quad {But}}\quad {\overset{\_}{\nabla}\quad {\times \overset{\_}{H}}}}}} = \overset{\_}{i}}$${{2{\pi\rho}\quad h\quad \theta} = {{\int{\int{{\overset{\_}{i} \cdot d}\overset{\_}{a}}}} = {I_{s} - \left( {{Im} + {Imc} + {Ixo}} \right)}}}\quad$

and It=Is−(Im+Imc+Ixo) since {overscore (∇)}·{overscore (i)}=0 and Isand (Im+Imc+Ixo) are in opposite directions. The formation resistance$r_{t} = {\frac{V_{s}}{I_{t}} = \frac{V_{s}}{2\quad \pi \quad \rho \quad h_{\theta}}}$

where I_(t) is measured from its effect on phase modulation and V_(s) isthe adjustable known supply voltage applied at the formation.

Here s refers to total supply electrode voltages and currents, m refersto mud, mc refers to mudcake, xo refers to transition zone, and t refersto the formation of interest surrounding the sensitive volume of themagnetic resonance well logging device.

Therefore, only current flowing axially but peripheral to the sensitivevolume will produce phase modulation of the spins within the sensitivevolume.

Further, whatever radial components of the total current traverse thesensitive volume do so in symmetrically opposite fields above and belowthe trans-axial plane of symmetry of the sensitive volume and produceh_(θ) fields that cancel.

APPENDIX 6 Signal to Noise Ration

Each sideband element is equally broadened by dephasing caused by staticvariations in the ambient Ho field, and is periodically rephased by theCPMG π pulse. Each sideband element can be represented by a pulse in thefrequency domain, or by a “sync” function in the time domain asillustrated in FIG. 13b.

The noise power limiting the estimation of each sideband signal strengthP_(n) is: P_(n) = K  τ  Δ  f ${\Delta \quad f} = \frac{1}{T}$

where K is Boltzman's constant, τ is the ambient temperature at thesignal generator (the formation), R₀ is the characteristic impedance ofthe transmission line, and Δf is the bandwidth of each sideband element,which element width is the same throughout the line spectrum.

Each sideband envelope can be written as:$v_{(t)} = {\hat{V}{\frac{T}{\pi} \cdot \frac{\sin \left( {\frac{\pi}{T}t} \right)}{t}}}$

where {circumflex over (V)} is a peak voltage set by the spin density,the sideband frequency (approximately the Larmor J₀( ) frequency), andby system geometry; and 2T is the width of the central lobe of the“sync” function. The Fourier transform F((ω) of V(t) is:

F(ω)={circumflex over (V)}TP _(π/T)(ω)

where P_(π/T)(ω) is a pulse function of width $\frac{2^{\pi}}{T}.$

By the Parseval-Bessel Theorem, the total energy E_(S) of v_((t)) is:$E_{s} = \frac{{\hat{V}}^{2}T}{4\quad R_{o}}$

If this energy E_(s) is assumed to lie almost entirely within the maincentral lobe of the “sync” function, the average signal power can beestimated as:${P_{S} \cong \frac{E_{s}}{2\quad T}} = \frac{{\hat{V}}^{2}}{8\quad R_{o}}$

Then, the signal-to-noise power ratio is:$\frac{P_{s}}{P_{n}} = {{\hat{V}}^{2}{T \cdot \frac{1}{8\quad K\quad \tau \quad R_{o}}}}$

For an ambient temperature of 400° Kelvin, in a 50 ohm line:$\frac{P_{s}}{P_{n}} = {{{\hat{V}}^{2}T\frac{1}{8\left( {1.38 \times 10^{- 23}} \right)(400)(50)}} = {{\hat{V}}^{2}{T\left( {{.453} \times 10^{18}} \right)}}}$

For a 4 ms pulse, T=2×10⁻³$\frac{P_{s}}{P_{n}} = {{\hat{V}}^{2}\left( {9.06 \times 10^{14}} \right)}$or$\hat{V} = {\left( \frac{P_{s}}{P_{n}} \right)^{1/2}\left( {0.333 \times 10^{- 7}} \right)}$

For${\frac{P_{s}}{P_{n}}100},{{\hat{V} \cong {0.3\quad \mu \quad V}};{20\quad {dB}\quad {signal}\text{/}{noise}}}$

For${\frac{P_{s}}{P_{n}}10},{{\hat{V} \cong {0.1\quad \mu \quad V}};{10\quad {dB}\quad {signal}\text{/}{noise}}}$

What is claimed is:
 1. An apparatus for determining a diffusioncoefficient of an object, comprising: a source of a first magnetic fieldalong a first axis to be applied to said object; a source of a secondmagnetic field along a second axis perpendicular to said first axis,where a frequency of said second magnetic field is equal to a Larmorfrequency specified by said object and said first magnetic field; anarray of conductors arranged substantially parallel to said firstmagnetic field direction; a voltage source which continuously orintermittently applies a voltage distribution to said array ofconductors, where a frequency of said voltage distribution is much lessthan the said Larmor frequency; and an element which cross-correlates areceived voltage distribution with said applied voltage distribution todetermine a diffusion effect as defined by said diffusion coefficient.2. The apparatus of claim 1, wherein: the source of the second magneticfield is a continuous or discontinuous source; said voltage distributionapplied to said set of conductors creates a time varying electric fieldin said object; said time varying electric field creates a time varyingcurrent density in said object; said time varying current densitycreates a time varying gradient field in said object; said time varyinggradient field dephases spins that diffuse and thereby change locationduring the application of said time varying gradient field; saiddephasing of spins varies a transverse magnetization in said object;said traverse magnetization induces said received voltage distributionin said array of conductors or in a second independent array ofconductors; and the diffusion of the spins defines a diffusioncoefficient which determines a relationship between said applied voltagedistribution and said received voltage distribution.
 3. A method fordetermining a complex permittivity of an object, comprising: applying afirst magnetic field along a first axis of said object; applying asecond magnetic field along a second axis perpendicular to said firstaxis, where a frequency of said second magnetic field is equal to aLarmor frequency specified by said object and said first magnetic field;continuously or intermittently applying a voltage distribution to anarray of conductors arranged substantially parallel to said firstmagnetic field direction, where a frequency of said voltage distributionis much less than the Larmor frequency; said voltage distributionapplied to said set of conductors creates a time varying electric fieldin said object; said time varying electric field creates a time varyingcurrent density in said object as determined by the complex permittivityof said object; said time varying current density creates a time varyinggradient field in said object; said time varying gradient field phasemodulates spins in said object, wherein said spins are nucleii innuclear magnetic resonance or unpaired electrons in electronparamagnetic resonance; said phase modulated spins in said object inducea received voltage distribution in said array of conductors or in asecond independent array of conductors; and cross-correlating thereceived voltage distribution with said applied voltage distribution todetermine said complex permittivity.
 4. An method for determining adiffusion coefficient of an object, comprising: applying a firstmagnetic field along a first axis of said object; applying a secondmagnetic field along a second axis perpendicular to said first axis,where a frequency of said second magnetic field is equal to a Larmorfrequency specified by said object and said first magnetic field;continuously or intermittently applying a voltage distribution to anarray of conductors arranged substantially parallel to said firstmagnetic field direction, where a frequency of said voltage distributionis much less than the said Larmor frequency; said voltage distributionapplied to said set of conductors creates a time varying electric fieldin said object; said time varying electric field creates a time varyingcurrent density in said object; said time varying current densitycreates a time varying gradient field in said object; said time varyinggradient field dephases spins that diffuse and thereby change locationduring the application of said time varying gradient field; saiddephasing of spins varies a transverse magnetization in said object;said traverse magnetization induces a received voltage distribution insaid array of conductors or in a second independent array of conductors;and cross-correlating said received voltage distribution with saidapplied voltage distribution to determine a diffusion effect as definedby said diffusion coefficient.